15 ISOLATED SINGULAR POINTS; POLES; ZEROS; RESIDUES 241 15. ISOLATED SINGULAR POINTS; POLES; ZEROS; RESIDUES (9 .15.1) Let A be an open subset of C, a an isolated point of C - A (3 .1 0.1 0), r a number >0 such that all points of the ball \z - a\ < r except a belong to A. Iff is an analytic mapping of A into a complex Banach space E, then for 0 < \z — a\ < r, we have n=0 n=l where both series are convergent for 0 < \z — a\ < r, and where y is the circuit t -* a + relt (0 < t ^ 2n). This follows at once from (9.14.2) applied to the ring p ^ \z - a\ < r, where p is arbitrarily small. 00 Observe that the series u(x) = £ dnxn is an entire function such that n = 1 w(0) = 0; we say that the function u(l/(z - a)) is the singular part of / in the neighborhood of a (or at a). When u = 0, / coincides in the open 00 set U: 0 < \z - a\ < r with the function g(z) = £ cn(z - a)n, which is n = 0 analytic for \z — a\< r; conversely, if/is the restriction to U of an analytic function/! defined for \z - a\ < r, then / =# by (9.9.4) and (9.15.1), hence u = 0. When u ^ 0, we say that # is an isolated singular point off. If u is a polynomial of degree n ^ 1, we say a is a />o/e of order n of/; if not (i.e. if dm ^ 0 for an infinite number of values of m) we say a is an essential singular point (or essential singularity) of/ In general, we define the order co(a;f) or co(a) of/at the point a as follows: co(0) = — oo if a is an essential singularity; 0)(a) = — n if a is a pole of order w > 1; o)(a) = m if/7^ 0, u = 0 00 and in the power series £ cn(z - 0)" equal to f(z) for 0 < |z - a\ < r, n = 0 m is the smallest integer for which cw^0; finally co(a; 0) = +00. When a>(a;f) ~ m > 0, we also say a is a zero of order m of/ Observe that if both /, g are analytic in the open set U : 0 < \z - a\ < r, and take their values in the same space, then co(a;f + g) ^ min(o>(0;/), o)(a;g)); if one of the functions/,^ is complex valued, then (o(a;fg) = a(a;fl + (a;g) is finite. Any function / analytic in U and of finite order n (positive or negative) can be written in a unique e Q and/e